A direct hit on a kindergarten in the southern Israeli city of Ashdod on Sunday afternoon caused considerable damage to the structure, but the kindergarten was empty at the time and no casualties were reported.
Earlier on Sunday, a rocket landed in Sderot, also in Southern Israel, damaging a house.
The rocket landed just 15-metres (50 feet) away from a school where students were studying.
It was not known if there were any injuries, though ambulances and emergency services were at the scene.
Sderot has been the frequent targets of rocket fire from the Gaza Strip and students there had only just returned to school after a two week absence due to security concerns.
In many other towns in southern Israel schools remained closed.
Rockets have struck several empty schools in the past two weeks.
Israel warned Gaza's 1.4 (m) million residents on Saturday that it plans to escalate a devastating air and ground assault that already has killed more than 800 Palestinians.
Israel launched its campaign on 27 December after years of Palestinian rocket attacks and has said many of the Palestinian civilian casualties are due to Hamas using civilian areas as hiding places and staging grounds for attacks.
Israeli defence officials say they are prepared for a third stage of their offensive, in which ground troops would push further into Gaza, but are waiting for approval from the government.
The first phase was a massive aerial bombardment, and the second saw ground forces enter Gaza, seize open areas used to fire rockets and surround Gaza City.
The officials, speaking on condition of anonymity because military plans have not been made public, said the army also has a contingency plan for a fourth phase - the full reoccupation of Gaza and toppling of Hamas.